Algebra 2 / Unit 2: Quadratic Functions and Complex Numbers
2.3 Imaginary and Complex Numbers
Notes
The imaginary unit is $\displaystyle i = \sqrt{-1}$, so $i^2 = -1$.
- For $a > 0$: $\displaystyle \sqrt{-a} = i\sqrt{a}$. Example: $\displaystyle \sqrt{-25} = 5i$.
- A complex number has the form $a + bi$: $a$ is the real part, $b$ is the imaginary part.
- Powers of $i$ repeat every 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. Divide the exponent by 4 and use the remainder.
- Now $x^2 = -9$ has solutions: $x = \pm 3i$.
Practice
Simplify $\displaystyle \sqrt{-49}$.
$\displaystyle \sqrt{-49} = i\sqrt{49} = 7i$.
Simplify $i^{23}$.
$23 = 4 \cdot 5 + 3$, so $i^{23} = i^3 = -i$.
Solve $x^2 + 16 = 0$.
$x^2 = -16$, so $\displaystyle x = \pm\sqrt{-16} = \pm 4i$.
Ready for more? Take the Unit 2 test.